We derive generalization error bounds for traditional time- series forecasting models. Our results hold for many standard forecasting tools including autoregressive models, moving average models, and, more generally, linear state-space models. These non-asymptotic bounds need only weak assumptions on the data-generating process, yet allow forecasters to select among competing models and to guarantee, with high probability, that their chosen model will perform well. We motivate our techniques with and apply them to standard economic and financial forecasting tools---a GARCH model for predicting equity volatility and a dynamic stochastic general equilibrium model (DSGE), the standard tool in macroeconomic forecasting. We demonstrate in particular how our techniques can aid forecasters and policy makers in choosing models which behave well under uncertainty and mis-specification.

We introduce a class of nonstationary covariance functions for Gaussian process (GP) regression. Nonstationary covariance functions allow the model to adapt to functions whose smoothness varies with the inputs. The class includes a nonstationary version of the Matérn stationary co- variance, in which the differentiability of the regression function is con- trolled by a parameter, freeing one from fixing the differentiability in advance. In experiments, the nonstationary GP regression model per- forms well when the input space is two or three dimensions, outperform- ing a neural network model and Bayesian free-knot spline models, and competitive with a Bayesian neural network, but is outperformed in one dimension by a state-of-the-art Bayesian free-knot spline model. The model readily generalizes to non-Gaussian data. Use of computational methods for speeding GP fitting may allow for implementation of the method on larger datasets.

One way to improve the efficiency of mine search, compared with a complete coverage algorithm, is to direct the search based on the spatial pattern of the minefield. This paper extends our original statistical approach (2001) to identify the regular pattern of a minefield at the beginning of the searching process. The extracted pattern parameters are used to build a probability distribution map of the configuration of the minefield. The map then can be used to guide the search for more mines efficiently. The new approach can efficiently capture the systematic and accumulated random departure of the actual mine locations from the grid pattern caused by the inaccuracy of the translational and rotational motion of the mine layer. Online implementation of our approach on a mobile robot is feasible.

One way to improve the efficiency of mine search, compared with a complete coverage algorithm, is to direct the search based on the spatial distribution of the minefield. The key for the success of this probabilistic approach is to efficiently extract the spatial distribution of the minefield during the process of the search. In our research, we assume that a minefield follows a regular pattern, which belongs to a family of known patterns. A Bayesian approach to the pattern extraction is developed to extract the underlying pattern of the minefield. The algorithm performs well in its ability to catch the "actual" pattern in the situation where placement and detector errors exist, and the algorithm is efficient, therefore, online implement of the algorithm on a mobile robot is possible. Compared to the likelihood approach, the advantage of using a Bayesian approach is that this approach provides information about the uncertainty of the extracted "actual" pattern.

We introduce two indices for the degree of incoherence in a set of lower and upper previsions: maximizing the rate of loss the incoherent bookmaker experiences in a Dutch Book, or maximizing the rate of profit the gambler achieves who makes Dutch Book against the incoherent bookmaker. We report how efficient bookmaking is achieved against these two indices in the case of incoherent previsions for events on a finite partition, and for incoherent previsions that include also a simple random variable. We relate the epsilon-contamination model to efficient bookmaking in the case of the rate of profit.

Questions of consensus among Bayesian investigators are discussed with regard to: (1) inference and (2) decisions. Concerning topic (1), results about asymptotic consensus and certainty with increasing evidence are reported. The findings deal with extending a conclusion of D. Blackwell and L. Dubins (1962) from pairs of agents to larger communities. Increasing evidence creates (almost sure) certainty of the truth and, depending upon the size of the community, it leads to varieties of consensus for conditional (posterior) distributions. Concerning topic (2), results are reported on the shared agreements of two Bayesian decision makers who have some differences in their probabilities for events and some differences in their utilities for outcomes. The results are couched in a setting where acts are so-called horse lotteries, with coherence defined by axiomatic restrictions on preferences over such acts. Subject to a weak Pareto condition, it is established that the only coherent preference schemes are the two agents' preferences themselves. Moreover, stronger Pareto conditions exclude even these extreme solutions.>